Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30
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The piecewise exponential flat function is not analytic at zero

Statement

False claim: the standard flat function is analytic at 0.

Facts & Assumptions

Given: The standard flat function β.

[F1]

For t>0, one has β(t)=exp⁡(−1/t)>0, while β(t)=0 for t≤0 (The standard flat function).

[L1]

Every derivative of β at 0 is equal to 0 (The standard flat function is smooth and flat at zero).

Refutation

technique · direct
1.1

By [L1], the Taylor series of β at 0 is the zero series.

L1
1.2

By [F1], the function β is positive on every punctured right neighbourhood of 0, so it is not equal near 0 to its zero Taylor series.

F1
2.1

Therefore β is not analytic at 0.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources